On the Relations between Fermion Masses and Isospin Couplings in the Microscopic Model

Abstract

Quark and lepton masses and mixings are considered in the framework of the microscopic model. The most general ansatz for the interactions among tetrons leads to a Hamiltonian $H_T$ involving Dzyaloshinskii-Moriya (DM), Heisenberg and torsional isospin forces. Diagonalization of the Hamiltonian provides for 24 eigenvalues which are identified as the quark and lepton masses. While the masses of the third and second family arise from DM and Heisenberg type of isospin interactions, light family masses are related to torsional interactions among tetrons. Neutrino masses turn out to be special in that they are given in terms of tiny isospin non-conserving DM, Heisenberg and torsional couplings. The approach not only leads to masses, but also allows to calculate the quark and lepton eigenstates, an issue, which is important for the determination of the CKM and PMNS mixing matrices. Compact expressions for the eigenfunctions of $H_T$ are given. The almost exact isospin conservation of the system dictates the form of the lepton states and makes them independent of all the couplings in $H_T$. Much in contrast, there is a strong dependence of the quark states on the coupling strengths, and a promising hierarchy between the quark families shows up.

Figures (3)

• Figure 1: The global ground state of the universe after the electroweak symmetry breaking has occurred, considered at Planck scale distances. It consists of an aligned system of tetrahedrons each extending into 3 extra dimensions. The big black double arrow represents 3-dimensional physical space. $a_L$ is the magnitude of one tetrahedron within the 3 extra dimensions and $a_P$ the average distance between two neighboring tetrahedrons. The small arrows are the isospin vectors $\vec{Q}_L+\vec{Q}_R$ defined in (\ref{['eq894']}) and used in (\ref{['mm3']}). Before the symmetry breaking the isospin vectors are directed randomly, thus exhibiting a local SU(2) symmetry, but once the temperature drops below the Fermi scale $v_F$, they become ordered into a repetitive tetrahedral structure, thereby spontaneously breaking the initial SU(2). Note that the SM Higgs vev $v_F$ is related to the length of the aligned isospin vectors in neighboring tetrahedrons. The figure is therefore a bit misleading, not only because the tetrahedrons do not have an extension into physical space, but also the relative magnitudes are not correctly drawn. While $a_L$ and $a_R$ are of the order of the Planck length, the extension of the tetrahedrons formed by the isospin vectors is dictated by the Fermi scale. While gravity can be attributed to the elasticity of the coordinate bondsbodogravity, the phenomena of particle physics arise from the interactions between isospin vectors. The figure shows how our universe looks like in the tetron model. It is part of a 6-dimensional space and is a 3-dimensional 'monolayer' of tetrahedrons each extending into the remaining 3 extra dimensions. The monolayer ground state acts as a background on which quarks and leptons glide as quasiparticle excitations. It has the properties of a Lorentz ether and is thereby not in conflict with Michelson-Morley type of experiments.
• Figure 2: Transition between the CKM and the PMNS limit of the matrix elements $|V_{12}|$ and $|V_{13}|$ and the Jarlskog invariant (from left to right) as a function of the parameter $\alpha$ defined in the main text. For example, $|V_{12}|$ starts with its CKM value 0.224 at $\alpha =0$ and grows towards the PMNS value at $\alpha =1$.
• Figure 3: The total isomagnetic potential V between 2 tetrons as a function of their distance. The Planck length $a_P$ is defined in Fig. 1 as the distance between 2 neighboring tetrahedrons, while $a_L$ corresponds to the extension of any one tetrahedron in the 3 extra dimensions. The minimum of V at $a_P$ essentially is due to the Heisenberg singlet potential $V_1$. $V_1$ is the dominant contribution to the inter-tetrahedral interactions and causes neighboring tetrahedrons of isospin vectors to align as shown in Fig. 1. On the other hand, the minimum at $a_L$ arises from the DM triplet potential $V_3$ introduced in (\ref{['eqrt15d']}). It dominates the inner-tetrahedral interactions and gives the top quark its mass in the sense that $m_t \sim K_{LL} \sim V_3(a_L)$, cf. (\ref{['all36']}). The values $V_3(a_L)$ and $V_1(a_P)$ are naturally of the same order, because they both stem from one original potential $W_7$ defined in connection with (\ref{['eq19']}) and (\ref{['eqrt15a']}). Note that $J_{inner} :=V_1(a_L)\sim 1$ GeV is much smaller than $J_{inter}:= V_1(a_P)\sim 100$ GeV and according to (\ref{['all36']}) is responsible for the mass of the second family.